= Solution
Choose a small skeleton $\mathcal F$ of the <finite spectra>, including every integer suspension. There is such a set of representatives: a finite list of stable cells and their attaching maps specifies a finite cell model, and the possible attaching maps form sets. For every $F\in\mathcal F$ and every stable map $v:F\to D$, take one copy of $F$ and form
$$
X=\bigvee_{(F,v)}F.
$$
Define the evaluation map $e:X\to D$ by taking its restriction to the $(F,v)$ summand to be $v$. Complete it to the <cofiber sequence of spectra>
$$
X\xrightarrow{e}D\xrightarrow{p}E\longrightarrow\Sigma X.
$$
This is the <universal evaluation phantom map> construction.
First, $p$ is nonzero. If $p=0$, applying $[D,-]_{\mathrm{st}}$ to the <cofiber sequence of spectra> gives the exact portion
$$
[D,X]_{\mathrm{st}}\xrightarrow{e_*}[D,D]_{\mathrm{st}}\xrightarrow{p_*}[D,E]_{\mathrm{st}}.
$$
Since $p_*(1_D)=p=0$, exactness supplies $s:D\to X$ with $es=1_D$. Thus $D$ is a <retract> of $X$. This is also a wedge summand: complete $s$ to a <cofiber sequence of spectra> $D\to X\to C'\to\Sigma D$. The left inverse $e$ makes this triangle split, giving $X\cong D\vee C'$ in the <stable homotopy category>. One can see the splitting directly from exactness: the connecting map is zero because $e$ extends $1_D$ across $s$, and consequently the quotient map $X\to C'$ admits a section. This contradicts the assumption on $D$. Hence $p\ne0$.
For any <finite spectrum> $F$ and any $v:F\to D$, its chosen representative has a summand inclusion $\iota_v:F\to X$ with $e\iota_v=v$, after transferring along the representative isomorphism. Consecutive maps of a <cofiber sequence of spectra> compose to zero, so
$$
pv=pe\iota_v=0.
$$
Thus $p$ is a <phantom map of spectra>. To establish the requested conclusion on all spaces, rather than just coefficient groups, let $K$ first be a finite based <CW complex>. Put $A=\Sigma^\infty K$. The <Spanier-Whitehead dual> $A^\vee=F(A,\mathbb S)$ is a <finite spectrum>. The evaluation and coevaluation adjunction of <Spanier-Whitehead duality> identifies
$$
\widetilde D_k(K)=\pi_k(D\wedge A)\cong[\Sigma^kA^\vee,D]_{\mathrm{st}}.
$$
Under this identification $p_*$ is postcomposition with $p$. Its source is a <finite spectrum>, including when $k$ is negative, so the calculation $pv=0$ proves that $p_*$ is zero on $\widetilde D_k(K)$ for every $k$.
Now let $Y$ be an arbitrary based <CW complex>. It is the filtered union, or filtered <homotopy colimit of spaces>, of its finite based subcomplexes $K$. The <suspension spectrum> functor and <smash product of spectra> preserve that homotopy colimit. The shifted <sphere spectrum> is compact, so taking <stable homotopy groups> gives
$$
\widetilde D_k(Y)=\operatorname*{colim}_{K\subset Y\ \mathrm{finite}}\widetilde D_k(K),\qquad\widetilde E_k(Y)=\operatorname*{colim}_{K\subset Y\ \mathrm{finite}}\widetilde E_k(K).
$$
Every element of the first group therefore comes from a finite $K$, where $p_*$ has just been proved zero. Consequently $p_*$ is zero on $Y$ as well. A based <CW complex> replacement extends the conclusion to arbitrary based spaces in the homotopy category. We have constructed
$$
\boxed{p:D\longrightarrow E,\qquad p\ne0\text{ stably},\qquad p_*:\widetilde D_*(Y)\longrightarrow\widetilde E_*(Y)\text{ is zero for every }Y.}
$$
This uses continuity of <represented homology theory>, not a continuity assertion for <represented cohomology theory>; the latter need not hold.
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